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Theorem bnj1212 28832
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1212.1  |-  B  =  { x  e.  A  |  ph }
bnj1212.2  |-  ( th  <->  ( ch  /\  x  e.  B  /\  ta )
)
Assertion
Ref Expression
bnj1212  |-  ( th 
->  x  e.  A
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ch( x)    th( x)    ta( x)    B( x)

Proof of Theorem bnj1212
StepHypRef Expression
1 bnj1212.1 . . 3  |-  B  =  { x  e.  A  |  ph }
21bnj21 28743 . 2  |-  B  C_  A
3 bnj1212.2 . . 3  |-  ( th  <->  ( ch  /\  x  e.  B  /\  ta )
)
43simp2bi 971 . 2  |-  ( th 
->  x  e.  B
)
52, 4bnj1213 28831 1  |-  ( th 
->  x  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547
This theorem is referenced by:  bnj1204  29042  bnj1296  29051  bnj1415  29068  bnj1421  29072  bnj1442  29079  bnj1452  29082  bnj1489  29086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-in 3159  df-ss 3166
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